Pricing Health Bene ts: A Cost-Minimization Approach

Transcription

1 Pricing Health Bene ts: A Cost-Minimization Approach Nolan H. Miller March 14, 2005 Abstract We study the role of health bene ts in an employer s compensation strategy, given the overall goal of minimizing total compensation cost (wages plus health-insurance cost). When employees health status is private information, the employer s basic bene t package consists of a base wage and a moderate health plan, with a generous plan available for an additional charge. We show that in setting the charge for the generous plan, a cost-minimizing employer should act as a monopolist who sells health plan upgrades to its workers, and we discuss ways tax policy can encourage e ciency under cost-minimization and alternative pricing rules. John F. Kennedy School of Government, Harvard University, Cambridge, MA, I thank Chris Avery, Pedro Barros, Suzanne Cooper, David Dranove, Bill Encinosa, Karen Eggleston, Je Liebman, Eduardo Loyo, Albert Ma, Tom McGuire, Joe Newhouse, Richard Zeckhauser, anonymous referees, and participants at the Boston University/Department of Veterans A airs Fifth Biennial Conference on the Industrial Organization of Health Care for helpful comments. Financial support of the Department of Veterans A airs is gratefully acknowledged.

2 1 Introduction In the United States, employer-provided health bene ts comprise an important part of employees total compensation. In 2003, approximately 69% of the population received private health insurance, and for 88% of them this insurance was employment based (DeNavas-Walt et al., 2004). The prevalence of employment-based health insurance is largely attributable to the cost of health insurance payments made through one s employer being exempt from taxation, and to employment-based group insurance being less expensive than individual (non-group) coverage due to lower per-capita administrative expenses and reduced adverse selection. Assuming workers value health insurance, these advantages make employer-provided health bene ts an e cient means of compensation. This paper analyzes how an employer should design and price its health bene ts when it cannot observe workers health care needs, given the goal of minimizing the total compensation cost of its workers. We consider an employer who o ers two plans, a moderate plan, such as an HMO, and a more generous plan, such as a PPO or indemnity plan. The employer s compensation scheme therefore consists of a base wage paid to all employees, and an additional surcharge imposed on those who elect the more generous health plan instead of the moderate one. 1 Although the employer may have an incentive to vary its base wage in order to in uence the makeup of its workforce, in this analysis we take the employer s workforce as given in order to focus on how the employer should set the surcharge for generous coverage. 2 The employer s costminimizing surcharge depends on several factors. When an employee elects generous coverage, he pays the surcharge to the employer, which e ectively decreases that employee s wage. However, since the generous plan is more expensive than the moderate one, this wage savings is partly o set by the additional cost of the employee s health insurance. In addition, as the employer increases the surcharge, fewer workers choose the generous plan, and this e ect must also factor into the employer s decision. The main insight of the cost-minimization approach is that in optimally balancing these e ects, the cost-minimizing employer should act as a monopolist who sells health plan upgrades to its 1 Use of the term surcharge is intended to emphasize that it is a charge in addition to the charge for the moderate plan. It is not intended to imply that this charge is unfair. 2 In particular, by decreasing its base wage, the employer lowers the wage paid to all of its employees but also increases its expected health care costs (because it attracts a sicker mix of workers). See Miller (2005) for an analysis of the employer s incentives to vary the base wage in order to a ect the composition of its workforce. 1

3 employees. The optimal program involves equating the appropriate concepts of marginal revenue and marginal cost. While the employer s monopoly power leads it to enroll too few workers in the generous plan (from a social perspective), the fact that the cost of employer-provided health bene ts is not taxable induces the employer to lower the charge for generous coverage and enroll more workers in the generous plan than it would if these expenditures were treated as taxable income to the workers. When health status is private information and the employer is unable to base an employee s wage on his health status, the bene t of the preferential tax treatment is shared between the employer and the workers. An important concern with employer-provided health bene ts is that an employer that o ers plans of di ering intensity exposes itself to adverse selection, which may jeopardize the plans viability. Given the choice of plans, those workers who expect to have the highest health care needs are drawn to the generous option. This adverse selection increases the average cost associated with that plan, leading the insurer to demand a higher per-worker premium from the employer. If the employer passes this price increase on to its employees, further adverse selection will result. Those in the generous plan with the lowest expected health care needs will choose to switch to the moderate plan, again increasing the average cost of caring for those who remain in the generous plan. In extreme cases, this phenomenon results in the so-called premium death spiral, with a high premium leading to adverse selection leading to an even higher premium, the process continuing until the generous plan is no longer sustainable. Adverse selection and premium spirals have received a great deal of attention in the literature, most often in contexts in which employers follow simple, non-optimal pricing rules. 3 This paper s analysis of cost-minimizing employers shows that these phenomena arise as consequences of the employer s approach to the bene ts-design problem, and that adopting the cost-minimizing approach will tend to decrease their severity. A sophisticated employer should recognize that self-selection will a ect the cost of any wage-bene t scheme and incorporate this into its planning. Seen in this light, the employer is not a passive force that may fall victim to adverse selection. Rather, it should design its compensation scheme in order to control selection and minimize its overall compensation 3 See Feldman and Dowd (1982, 1991), Cutler and Zeckhauser (1998), Cutler and Reber (1998), and Ellis and Aragao (2001). 2

4 cost. This paper shows that as long as there are some workers for whom the incremental bene t of the generous plan is greater than its incremental cost, the generous plan is viable (i.e., it is chosen by a positive mass of workers) when the employer prices it according to the cost-minimization rule. 4 Thus, the cost-minimizing employer o ers the generous plan whenever it is socially bene cial to do so. Other rules, such as the commonly-employed equal lump-sum contribution rule (in which the employer pays a xed dollar amount toward the employee s health insurance regardless of which plan he chooses) do not have this property. It may be that o ering the generous plan is both socially e cient and cost e ective for the employer, yet employers following the equal lump-sum contribution rule fail to o er a viable generous plan. This paper adopts a total compensation view of health bene ts, according to which workers employment decisions are driven by the total value of the compensation package o ered to them, i.e., both wages and insurance contribute to inducing the worker to accept employment. Although we believe our implementation of the idea is novel, the view itself is not new. Pauly (1997) espouses this approach, as does the labor economics literature on compensating di erentials, summarized in Rosen (1986), which holds that in a competitive market relative wages adjust so that a worker in a particular job is just compensated for any extraordinary bene ts (or costs) he receives. A fundamental di erence between this paper and previous academic work on health bene ts is in the explicit treatment of employers as acting strategically in order to minimize expected compensation cost. 5 This is seen most clearly when the present paper is compared with Cutler and Reber (1998, hereafter CR), which employs an almost identical model to study the e ects of Harvard University s change in its premium contribution strategy on employee demand for its plan o erings and on plan competition. Because the analysis is descriptive, CR does not take a position on Harvard s objective in designing its bene ts package; it merely documents the e ects of the rule change. Focusing on the adverse selection part of the problem, this paper provides a theory of 4 Although the cost-minimizing scheme o ers the generous plan to some workers, it need not o er it to all workers for whom the marginal bene t outweighs the marginal cost. 5 Two notable exceptions are Levy (1998), which examines the possibility that employers might use employee contributions for health plans as a means of separating workers who want health insurance from those who don t, allowing the employer to avoid buying health care coverage for those who do not value it, and Dranove, Spier, and Baker (2000), which argues that employers may require employees to contribute toward the cost of their health insurance in order to induce married employees to acquire health insurance from their spouses employers instead. 3

5 how, from its own point of view, a cost-minimizing employer should act. The paper continues with a description of the model in Section 2. Section 3 discusses costminimization with full information, and Section 4 derives and analyzes cost-minimizing compensation programs with private information. Section 5 discusses commonly used pricing rules, and Section 6 concludes. 2 The Model 2.1 Basic Structure The basic structure of the model is similar to CR. The employer s workforce is xed. Employees di er in expected health care costs but are otherwise identical. Let c 2 [0; ], > 0, denote a worker s type, where c is normalized to be the expected cost of health care for this worker if enrolled in the generous plan. The cost of care for a worker of type c if enrolled in the moderate plan is given by c, where 2 (0; 1). The consensus in the literature is the is somewhere between 0:8 and 0:9 in the case where the moderate plan is an HMO and the generous plan is a PPO. 6 An employee s utility is the sum of his wage and the utility from his health plan. We assume that all employees receive some health plan. 7 Denote the dollar-valued bene t derived by a worker of type c enrolled in the moderate plan as m (c), where m 0 (c) > 0 for c 2 (0; ) : Let g (c) denote the additional bene t derived by enrolling in the generous plan, where g (c) is strictly increasing and strictly convex (i.e., g 0 (c) > 0 and g 00 (c) > 0 for c 2 (0; )), and g (0) = 0. 8 Thus, a worker receiving (after-tax) wage w has utility w + m (c)if he is enrolled in the moderate plan and w + m (c) + g (c) if he is enrolled in the generous plan. We assume that all workers have (after-tax) reservation utility u 0, which is independent of c. 9 The employer is risk neutral, and it is only concerned with minimizing its expected compensation 6 The 0:9 gure is put forth in Cutler and Reber (1998). Compared to an HMO, the Congressional Research Service reports is approximately 0:82 when the generous plan is a fee-for-service plan, 0:89 when the generous plan is a preferred provider organization, and 0:88 when the generous plan is a point-of-service plan (Congressional Research Service, 1997). 7 Miller (2005) considers when it is in the employer s interest to o er its workers a no-insurance option. 8 Convexity ensures that a type-c worker never prefers a less generous plan than a lower-type worker. 9 This would be the case if, for example, workers who are not employed by the rm do not purchase health insurance on their own. See Miller (2005) for an analysis of the case where workers outside option includes the opportunity to purchase a health plan from the non-group market. 4

6 cost. While each worker knows his type, for the bulk of the analysis we assume that the employer is unable to observe workers types. Let F (c) be the cumulative distribution function of types in the employer s workforce, where F (0) = 0; F () = 1. Distribution F (), assumed known by the employer, has continuous density function f (c). 2.2 The Insurance Market The employer contracts with an insurer to provide health insurance for its employees. This paper makes two assumptions regarding insurance providers. First, we assume that, based on observable characteristics of workers such as age, sex, and current health status, the insurance company is able to generate an unbiased estimate of the cost of caring for any individual worker (or group of workers). 10 Because the insurer is risk neutral, it is e ectively as if the insurer can observe each worker s expected cost of care, i.e., his type. Second, we assume that insurance companies operate in a perfectly competitive insurance industry. This assumption implies that, in equilibrium, the expected pro t earned on any plan must be zero. Thus, for example, if the insurer provides the employer s generous plan, the total premium paid to the insurer in any equilibrium must equal the expected cost of caring for those enrolled in the generous plan. While the zero-pro t condition implied by the perfect-competition assumption appears strong, it is a necessary requirement for any equilibrium, given perfect competition in the insurance market. An alternative formulation of the insurance industry s breakeven constraint might require that the insurer break even overall but need not break even on each individual plan. That is, total payments to the insurer must equal the expected cost of care across those enrolled in both plans, but there may be cross-subsidization between the plans. However, such cross subsidization seems inconsistent with perfect competition. When the moderate plan subsidizes the generous plan, one might expect another insurer to o er the moderate plan at a lower cost This assumption is consistent with industry practice, where insurers price policies based on an assessment of health risk using demographic data, but could also involve physical examinations of the members of the group. See Pauly and Herring (1999) for a discussion of the underwriting of individual and group policies. 11 Further, as will become clear, our cost-minimizing employer is concerned with its total compensation cost, not how this cost is divided among those receiving the moderate and generous plans. Any insurer who is willing o cross-subsidize its plans will do so only if the employer purchases both plans from it. However, in this case crosssubsidization does not a ect the total cost of health insurance remains the same. Thus, as long as the insurer earns zero pro t overall, whether or not it cross-subsidizes has no a ect on the employer s optimal strategy. 5

7 Before considering the employer s decision, we rst characterize the socially optimal allocation of workers to plans. The incremental cost of enrolling a type-c worker in the generous plan is (1 ) c, and the incremental bene t is g (c). Hence, the surplus-maximizing allocation of workers to plans is for a type-c worker to elect generous coverage if and only if g (c) (1 ) c. To focus in the interesting case where each plan is e ciently provided to some workers, we assume that: g 0 (0) < 1 and g () > (1 ), (1) which implies that there exists a unique worker type c E 2 (0; ) such that: g (c E ) = (1 ) c E. (2) That is, for type c E the marginal bene t from the generous plan just equal its marginal cost. Under the e cient allocation, workers of type 0 c < c E receive the moderate plan and workers of type c E c receive the generous plan, where without loss of generality we adopt the convention that workers indi erent between the two plans choose the generous one. 2.3 Taxability of Employer Provided Health Bene ts In the United States, the cost of employer-provided health insurance is not considered taxable income to the employee for either federal or state income taxes, nor is it included in the tax base for (social security) payroll taxes. 12 Let t (with 0 < t < 1) denote the relevant individual tax rate, inclusive of all of these factors. For the purposes of this analysis, we ignore the question of whether the employer requires workers electing the moderate plan to pay part of the cost of their health insurance. Employees care only about their net wage and the health plan they receive, and under Section 125 of the Internal Revenue Code an employee s contribution for health insurance made through his employer can be made tax deductible (Gruber 2000). Thus, each dollar the employee must contribute for the moderate plan reduces the worker s net after-tax wage by 1 the worker, the employer must increase his after-tax wage by 1 t dollars. In order to compensate t dollars, which costs the employer 12 See Gruber and Poterba (1996). We ignore the corporate tax since both wages and health care expenditures are tax deductible for corporations. 6

8 1 dollar. Thus, whether the employee contributes for the moderate plan does not a ect the employer s overall cost. To simplify the analysis, we assume that the employer takes advantage of Section 125. The results change only slightly if it does not. 3 Cost Minimization with Full Information Before considering the employer s problem when workers types are unobservable, we rst brie y discuss the employer s problem when it can observe the health status of potential employees and base its compensation package on this information. Under full information, the cost-minimizing employer chooses a type-speci c wage for each worker so that the worker earns exactly his reservation utility from employment. This implies that workers receiving moderate coverage are paid pre-tax wage u 0 1 t and workers receiving generous coverage are paid u 0 m(c) g(c) 1 t. Since a type-c employee costs the employer u 0 m(c) 1 t + c if he receives moderate coverage and u 0 m(c) g(c) 1 t + c if he receives generous coverage, the employer prefers that the type-c worker receive generous coverage whenever: m(c) g (c) (1 t) (1 ) c. (3) Let c F be the lowest-cost employee for which the employer prefers generous coverage to moderate, i.e., c F satis es (3) with equality. Comparing (3) with (2) shows that c F < c E. That is, with full information the employer gives generous coverage to some employees for whom the incremental bene t is less than its incremental cost. The reason for this is that, due to the tax advantage a orded employer-provided health bene ts, the employer s cost of providing generous coverage is less than the true cost. Hence, c F can also be thought of as the cut-o point for the tax-preferred socially optimal allocation of workers to plans, i.e., treating the employer s tax-preferred cost as the true cost of care. In the full information case, the entire bene t of treating the employer s expenditure on health insurance as non-taxable accrues to the employer. To see why, note that as long as the employer can o er each worker a type-speci c wage (as it can in the full information case), every type of employee will receive exactly his reservation utility from employment. This is true regardless of the 7

9 taxability of the cost of health insurance. While making expenditures on health insurance taxable will a ect which employees are enrolled in which plan, any bene ts the workers receive from this will be o set by reduced wages Cost Minimization with Private Information While the full information case provides a useful benchmark, in practice employers rarely base what they charge employees for health insurance on health status, either because this information is unavailable to employers or because the employer is unable (or unwilling) to use it for compensation purposes. 14 Because of this, the remainder of the analysis is concerned with the private information case in which the employer does not base the compensation package it o ers to a particular worker on that worker s health status. If the employer either does not know workers types or is unable to act upon this knowledge, then its compensation plan will consist of a choice between moderate coverage and a higher wage or generous coverage and a lower wage. Thus, the di erence in the wages can be thought of as the surcharge imposed on those who choose the generous plan. The employer s task is then to choose wages for workers electing each health plan (i.e., the surcharge) in order to minimize the expected compensation cost of its workers, subject to the constraints that each worker receives at least his reservation utility and chooses the health plan that maximizes his net bene t from employment. 4.1 Equilibrium Requirements In our model, three parties make decisions: the employer, the workers, and the insurers. Our chief interest is in understanding the in uence of the employer s compensation scheme on the resulting equilibrium and its behavior. Consequently, we impose the following structure on the game. The employer moves rst, choosing its compensation practice. Then, after observing the employer s wage o erings, workers and insurers simultaneously choose their strategies. 13 In fact, if expenditures on health insurance are treated as taxable income, the employer will assign workers to plans in an e cient manner, i.e., workers of type 0 < c < c E will be given the moderate plan and workers of type c E c will be given the generous plan. 14 Encinosa and Selden (2001) report that in 1993 only 0.6% of workers receiving employment-related single coverage faced contributions that depended directly on health status, and less than 5% faced contributions that depended on anything at all, including smoking status, gender, age, or income. 8

10 Since the workers and insurers observe the employer s action before choosing their own strategies, the equilibrium concept we employ is subgame perfect Nash equilibrium. The equilibrium requirements (ER) are: ER1: The employer chooses a menu of wage-health plan pairs in order to minimize its expected compensation cost (including both wage and health insurance costs) given that workers and insurers react optimally to this compensation scheme. ER2a: Given the employer s compensation scheme, each worker chooses the insurance/wage pair that maximizes his utility (and satis es his reservation utility requirement). ER2b: Given the employer s compensation scheme, the insurers compete among themselves to o er insurance plans to the employer. All plans o ered in equilibrium must earn zero pro t. Because insurers and workers move simultaneously, equilibrium requirements ER2a and ER2b are imposed at the same time and de ne the equilibrium of the subgame created by the employer s choice of compensation scheme. Condition ER1 then requires that the employer choose the compensation plan that induces its most preferred subgame equilibrium. Equilibrium requirements ER2a and ER2b are the same as those imposed by CR, except we impose the additional requirement that each worker must receive at least his reservation utility from employment. As in CR, in the present paper the employer moves rst and the insurers and employees react to its choice. However, there is no analog to ER1 in CR because that paper does not posit that the employer chooses its compensation plan in order to maximize some objective. CR simply takes the employer s choice (i.e., Harvard s change in its premium-subisidy policy) as given and derives the second-stage equilibrium. 4.2 Worker Behavior Since we have taken the employer s workforce as xed, in order to meet workers reservation utility requirements, the employer must o er (after-tax) wage w 0 = u 0 m (0) to employees who receive the moderate health plan. Since the bene ts from health insurance increase with c, this wage ensures that all worker types earn a surplus of at least u 0 from employment. Let w g denote the after-tax take-home wage paid to workers who elect generous insurance. It is often convenient to 9

11 think of w 0 as the base wage o ered to all employees and w 0 w g as the surcharge to employees for generous coverage, measured in terms of the reduction in after-tax wages. Without loss of generality, assume w g w 0. Since we do not consider other forms of employee cost sharing (e.g., copayments or deductibles), the surcharge is equivalent to the employee s out-of-pocket contribution for the generous plan. A worker who chooses the moderate plan earns utility w 0 +m (c), and a worker who chooses the generous plan earns utility w g + m (c) + g (c). Hence, a type-c worker prefers generous coverage if and only if g (c) w 0 w g. Since g (0) = 0 and g (c) is increasing, de ne c g to be the lowest-cost worker who elects generous coverage. That is, g (c g ) = w 0 w g, (4) if such a c g exists. 15 Given wages w 0 and w g, the workers who accept employment self-select into two groups. Workers of type 0 c < c g choose moderate coverage at wage w 0, and workers of type c g c choose to pay the surcharge and receive generous coverage at wage w g. When workers types are private information, all workers except those of type c = 0 earn more than their reservation utility from employment. Workers of type 0 < c < c g earn utility w 0 +m (c), while workers of type c g c earn utility w 0 + m (c) + g (c) g (c g ). Thus, not only do almost all worker types earn a rent, but this rent is increasing in workers health care needs. This is an example of the informational rent earned by less attractive agents in any pooling equilibrium. Unhealthy workers bene t from the presence of healthy workers because high wages must be o ered to all in order to induce those who get little bene t from health care to accept employment. 4.3 The Employer s Optimization Problem We now turn to the employer s cost minimization problem. If workers of type 0 c < c g choose moderate coverage and workers of type c g c choose generous coverage, perfect competition among insurers implies that, in equilibrium, the employer s cost of the moderate and generous plans are R c g 0 cf (cjc m) and R c g cf (cjc m ) dc, respectively. 16 Therefore, anticipating equilibrium behavior 15 If w 0 = w g, then c g = 0. If w 0 w g > g (), then c g =. 16 CR assume the same cost structure. This is also the relevant cost structure if the employer self-insures, as do many large employers. Marquis and Long (1999) analyze self-insurance in seven states and nd that 56% of employers 10

12 by the workers and insurers, the employer s cost minimization problem is: Z cg Z 1 1 min 0w gw 0 0 (1 t) w m + c f (c) dc + c g (1 t) w g + c f (c) dc; (5) subject to (4). Since (4) de nes a one-to-one relationship between w g and c g, we can treat the employer s problem as choosing c g, and rewrite the employer s problem as: Z cg Z min (c g) = cf (c) dc + cf (c) dc (6) 0c g 0 c g + 1 (1 t) w 0 1 (1 t) g (c g) Z c g f (c) dc: The rst two terms of (c g ) comprise the expected cost of health care, the third term is the cost of the base wage paid to all workers, and the nal term is the portion of wages recovered as surcharges for the generous health plan. Throughout the paper, we assume a unique solution to the employer s problem, which we denote c g. Di erentiating (6) with respect to c g yields the following (necessary) characterization of c g. Proposition 1 : The cost-minimizing choice of c g is characterized by: 1 (1 t) g0 c g 1 F c 1 g + (1 t) g c g 8 (1 ) c g f c >< g >: 0 if c g = 0 = 0 if 0 < c g < 0 if c g =. (7) The surcharge for generous coverage is w 0 w g = g c g. Proof: Follows directly from the rst-order condition for (6). Consider an interior solution, i.e., one where (7) holds with equality. The rst term on the lefthand side of (7) captures the fact that increasing c g increases the surcharge that can be imposed on all workers who elect generous coverage, since the size of the surcharge is determined by the marginal bene t of generous coverage earned by the lowest-cost employee who elects it. Increasing c g also with more than 500 employees self-insured in

13 causes some workers who used to elect generous coverage to elect moderate coverage instead. These workers must then be paid a higher wage but incur lower expected health costs. The di erence in these two, the impact on total compensation of shifting the marginal worker to the moderate plan, is captured by the second term on the left-hand side of (7). Under the cost-minimizing plan the surcharge for electing generous coverage is g c g. One way to interpret this is that the employer is a monopolist who sells health care plans to its workers at price g c g. To see why, let p = w0 w g be the price of generous insurance. Demand for generous health coverage is then given by D (p) = 1 F g 1 (p). The employer s insurance cost is: Z g C (p) = 0 1 (p) Z cf (c) dc + cf (c) dc: g 1 (p) Hence for xed w m the employer s problem is equivalently written: max p0 1 D (p) p C (p). (1 t) Di erentiating with respect to p and setting the result equal to zero, 1 1 F g 1 (~p) f g 1 (~p) dg 1 (~p) ~p (1 t) dp + (1 ) dg 1 (~p) g 1 (~p) f g 1 (~p) = 0; dp at the optimal price, ~p. Substituting in that dg Let ~c g be the lowest-cost type who elects generous coverage at price ~p. 1 (p) dp = 1 g 0 (c g) and g 1 (p) = c g, and rearranging, 1 (1 t) g0 (~c g ) (1 F (~c g )) + 1 (1 t) g (~c g) f (~c g ) (1 ) ~c g f (~c g ) = 0 (8) Note that (8) is identical to (7), and therefore ~c g = c g. Seen this way, g c g is the price paid by each worker who elects generous coverage, and (7) is the monopolist s optimality condition requiring that, at the optimum, the marginal revenue due to selling the generous plan to slightly sicker people (the rst two terms of (8)) is just equal to the marginal cost of providing them with more generous care (the nal term of (8)). For ease of exposition, we have assumed that c g is unique. However, in light of the monopoly analogy, it is straightforward to see that the problem of multiple solutions for the cost-minimizing 12

14 employer is the same as that facing a monopolist with a non-concave pro t function. In this case, condition (7) is necessary but not su cient for a global maximum. Investigation of the second order conditions and comparison of local maximizers will identify the global maximum. 17 The corner solution where c g = 0 is easily ruled out under our assumptions. Corollary 1 :The cost-minimizing employer o ers moderate coverage to at least some employees, i.e., c g > 0. Proof: At c g = 0, the second term on the left-hand side of (7) is zero since g (0) = 0. The rst term is negative, contradicting the possibility that c g = 0 is optimal. Intuitively, workers with low health-care needs assign a vanishingly low value to generous coverage, and therefore if the employer wants them to choose generous coverage it must select a very small surcharge. However, since this means the surcharge must be low for all workers, it is preferable for the employer to increase the surcharge, just as it is in a monopolist s interest to raise the price it charges above the competitive price. The other corner solution, where c g =, is of particular importance because it represents the situation where the forces of adverse selection are so strong that the employer does not nd it worthwhile to o er the generous plan. The possibility that the cost-minimizing employer could prefer not to o er a viable generous plan can also be eliminated. Evaluating (7) at c g =, this situation is possible only if: g () (1 t) (1 ). (9) However, condition (9) can be satis ed only if it would not be socially desirable to o er the generous plan to any worker, i.e., only if no worker values the generous plan more than its incremental cost. This possibility is ruled out by assumption (1). Corollary 2 : The cost-minimizing employer o ers generous coverage to at least some employees, i.e., c g <. Proof: According to assumption (1), g () > (1 ), and so g () > (1 t) (1 ), contradicting the possibility that c g = is optimal. 17 For a wide variety of natural distributions of c and speci cations of g (c), the solution of the employer s problem is, in fact, unique. 13

15 While Corollary 2 establishes that the employer will give generous coverage to some employees, it does not imply that the employer s plan will give generous coverage to all workers for whom it is worthwhile, i.e., that c g = c E. As noted above, the employer s incentive to act as a monopolist will give it a tendency to restrict the quantity of workers receiving generous coverage in order to drive up the price. To gain further insight into the employer s problem, it is useful to rewrite (7) as (for an interior solution): g c g = (1 t) (1 ) c " g, (10) where " is the elasticity of demand (willingness to pay) for the generous plan, " = f(c g) g(c g) (1 F(c g)). The left-hand side of (10) is the monopolist s marginal revenue. Comparing (10) with (3) establishes that fewer workers receive generous coverage under private information than under full information, i.e., c g > c F. Extending the monopoly analogy, in the full information case, the employer is a perfectly price discriminating monopolist, reducing the wage of g 0 (c g) each worker who receives generous coverage by his willingness to pay for it. Because of this, the employer has an incentive to o er generous coverage to all workers who value generous coverage more than its (tax-subsidized) incremental cost. it charges a price above the competitive price. When the monopolist cannot price discriminate, The result is that fewer workers receive generous coverage. Expression (10) is useful in thinking about how the preferential tax treatment a orded employerprovided health bene ts impacts employer policy and through it employee welfare. Even when there is no tax advantage to providing health bene ts, the employer still has an incentive to act as a monopolist, which results in the employer enrolling fewer workers in the generous plan than is socially optimal. That is, if t = 0, c g > c E. Relative to this benchmark, making employer-provided health bene ts tax advantaged decreases the rm s marginal cost of providing generous coverage to more workers, and therefore induces the employer to charge less for the generous plan and provide generous coverage to more workers. 18 Thus, while the employer s monopoly power leads it to charge a high price for generous coverage and enroll too few workers (from a social perspective) in the 18 Selden (1999) proves the related result that in a Rothschild-Stiglitz model a linear premium subsidy (such as preferential tax treatment) reduces adverse selection. 14

16 generous plan, the tax deductibility of employer-provided health insurance reduces this distortion. Indeed, it is straightforward to show that there exists a tax rate that induces socially optimal sorting. Unlike in the full information case, when worker types are private information (or the employer is unable/unwilling to use this information in settings wages), the rm and workers share the bene ts of the tax subsidy. As in the full information case, the rm bene ts directly through the fact that the government now subsidizes the cost of its health plans. On the other hand, since the employer cannot hold workers to their reservation utility levels when it cannot base wages on health status, some of the bene t of the subsidy passes through to the workers. In particular, the workers who received generous care in the absence of the tax subsidy bene t through the lower charge, and the additional workers who receive generous care only when health insurance expenditures are subsidized bene t because these workers value generous care more than its out-of-pocket cost, g (c g ). 19 The only workers who do not bene t from the tax subsidy are those who receive the moderate plan both with and without the tax subsidy. Expression (10) also suggests that as employee demand for employer-provided generous coverage becomes more elastic, the employer provides generous coverage to more of its employees at a lower price. In fact, as demand becomes perfectly elastic (i.e., "! 1 ), the employer s choice of c g approaches the tax-preferred socially optimal level, where g (c) = (1 t) (1 ) c. Speculating outside the model considered here, factors that might make demand for employer-provided generous coverage more elastic include provision of better substitutes in the form of government-provided or subsidized individual coverage, decreases in barriers to switching jobs, or increased employer competition for employees Other Rules and Adverse Selection The cost-minimization approach focuses on setting the surcharge for the generous plan to induce the employees to sort themselves as the employer desires. The size of the surcharge determines 19 Increasing the rate of tax-preference also reduces the cost of employing unhealthy workers, which provides a counterbalance to incentives for health-based discrimination. I thank a referee for this observation. 20 In a Rothschild-Stiglitz model, Crocker and Moran (2003) study the role of barriers to worker mobility in promoting more complete insurance. 15

17 how employees will sort themselves into plans and the equilibrium premiums the plans will charge. Given this surcharge, the employer must pay the remainder of the employees premiums. Thus, the surcharge and employees and insurers equilibrium behavior determine the employer s subsidies. Firms actual approaches to pricing their health plans typically involve the opposite order. Employers begin by assuming that each worker should pay the per-worker premium that the insurer charges the employer for the plan that employee chooses. Taking this as a baseline, the employer then decides how and in what way to subsidize the employees insurance cost. Employees and insurers equilibrium behavior combine with the employer s subsidy policy to determine the insurance premiums and, through them, the di erence in the employees out-of-pocket contribution for the generous plan, i.e., the surcharge. 21 The two most widely used rules have the employer either paying an equal lump sum subsidy (ELS) for the employee s insurance, regardless of which plan is chosen, or the employer subsidizing an equal percentage of the premium (EPP) of whichever plan the employee chooses. 22 In either case, the worker bears the remainder of the premium cost for whichever plan he elects. Since the premium competitive insurers charge employers for a particular plan is proportional to the average cost of its enrollees, under either the ELS or EPP subsidy rule the cost to the worker of changing plans depends on enrollment choices of other workers. This dependence leaves the system vulnerable to adverse selection and, in extreme cases, the type of premium death spiral experienced by Harvard University in the mid 1990s, during which Harvard s most generous health plan became so expensive that Harvard stopped o ering it. CR studies the Harvard case. In the early 1990s, Harvard s health plan o erings included the generous Blue Cross/Blue Shield PPO as well as several less-generous HMO s. In 1995, the university changed its premium-contribution policy from a generous EPP-subsidy rule to an ELS- 21 The di erence in the order in which quantities are determined is apparent in CR. In order to implement the e cient assignment of workers to plans, they suggest choosing di erent lump-sum subsidies for the plans. In this case, the workers out-of-pocket contribution for the generous plan is the sum of the di erence in premiums and the di erence in subsidies. Appropriate choices of the unequal subsidies will, in fact, induce e cient sorting, although computation of the subsidies needed to do so is somewhat complicated. The task is much simpler for an employer following the approach of this paper, who sets the surcharge in order to manage worker self-selection. If such an employer had the goal of inducing e cient sorting (rather than minimizing cost), it would simply set the out-of-pocket contribution for the generous plan at g (c E) ;which would lead to the e cient allocation of workers to plans. 22 According to the Kaiser/HRET Employer Health Bene ts Annual Survey (2002), 17% of workers had employers followed an ELS rule in 2002 (down from 28% in 1999) and 37% of workers had employers who followed an EPP rule in 2002 (up from 29% in 1999). Encinosa and Selden (2001) provide a discussion of the use of risk adjustment and other methods of pricing employee health bene ts. 16

18 subsidy rule which imposed on workers a larger portion of the incremental cost of more generous care. As workers switched from the PPO to the HMOs, the di erence in the average cost of the two plans increased. Under the ELS subsidy rule, this caused the employees out-of-pocket contribution for the PPO to increase as well, which lead to further adverse selection. In just three years, declining enrollment forced the most generous plan, the Blue Cross/Blue Shield PPO, to be withdrawn because it could no longer be o ered at a reasonable price. Under an ELS subsidy rule such as the one Harvard used, the price paid by a worker who moves from the moderate plan to the generous plan is given by the di erence in the average cost of those enrolled in each of the plans. This price, which we denote p (c g ), is given by: 23 p (c g ) = Z c g f (c) c 1 F (c g ) dc Z cg 0 c f (c) F (c g ) dc. Since such expenditures are made with pre-tax dollars, the worker s after-tax cost of the generous plan is (1 t) p (c g ). Increasing c g increases the average cost of both the generous and moderate plans. Consequently, whether p (c g ) increases or decreases generally depends on the distribution of health costs. To simplify the discussion, we make the standard assumption that p (c g ) crosses g (c g ) no more than once. The equilibrium under the ELS rule involves a type-c worker choosing generous coverage if and only if g (c) > (1 t) p (c g ). Given that p (c g ) and g (c g ) cross no more than once, the equilibrium under the ELS rule involves workers of type 0 c < c ELS g of type c ELS g c electing generous coverage, where c ELS g if such a point exists. If no such point exists, it must be that g (c g ) > (1 electing moderate coverage and workers satis es g c ELS g = (1 t) p c ELS g, this case, no workers elect the generous plan in the ELS equilibrium, and we let c ELS g loss of generality. t) p (c g ) for all c g > 0. In = without Under the ELS rule, the employee s price for the generous plan is always greater than the true incremental resource cost of the marginal employee: p (c g ) > (1 ) c g. This is the fundamental reason why the ELS rule promotes adverse selection. Thus, in the absence of the tax-advantaged status of expenditures on employer-provided health bene ts, the ELS rule enrolls too few workers in the generous plan: c ELS g > c E. By decreasing the out-of-pocket contribution for the generous 23 Cutler and Reber (1998, p. 437) denote this quantity the premium-out-of-pocket, or P oop. 17

19 plan, the tax advantage induces more workers to choose the generous plan. As with the costminimization rule, there is a tax rate that induces socially optimal sorting. Under the ELS rule, the generous plan is not viable (i.e., is chosen by no workers) whenever g (c g ) < (1 t) p (c g ) for all c g. This is likely to be the case when workers incremental value of the generous plan is small, the tax rate is low, or p (c g ) is uniformly large. 24 As noted in (9) above, under the cost-minimizing scheme the generous plan remains viable whenever g () > (1 t) (1 ). Comparing the ELS and cost-minimizing rules leads directly to Proposition 2. Proposition 2 : If the generous plan is viable under the ELS subsidy rule, then it is viable under cost minimization as well. However, the converse is not true. It may be that an employer following the cost-minimization rule would provide the generous plan to some workers, but an employer following the ELS rule would not. Proof: Z Z! f (c) cg (1 t) p (c g ) = (1 t) c c g 1 F (c g ) dc f (c) c c F (c m g ) F (c m) dc Z! cg f (c) > (1 t) c c F (c m g ) F (c m) dc > (1 t) (1 ) : Hence if (1 t) p (c g ) < g (c g ) for some c g, then (1 t) (1 ) < g (), proving the rst part of the proposition. But, the reverse implication does not hold, i.e., it is possible to nd speci cations of the parameters of the problem where max cg f(1 t) p (c g )g > g () > (1 t) (1 ) without violating any of our other assumptions, which establishes the second part of the proposition. Proposition 2 establishes that there are times when an employer subsidizing according to the ELS rule would lose its generous plan to a premium death spiral, whereas the generous plan would have continued to be o ered had the employer instead followed the cost-minimization approach. This is most likely to occur when p (c g ) is uniformly large, i.e., when there is substantial dispersion 24 If f (c) is highly concentrated near a single point, then p (c g) will tend to be small. When the distribution does not have this feature (e.g., the distribution has fat tails or multiple points of high density that are far from each other), p (c g) is more likely to be uniformly large. 18

20 in the distribution of health types. 25 In such cases the employer s total compensation cost is lower when the generous plan is o ered than when it is not. Thus, under the ELS-subsidy rule, there are cases in which the death of the generous plan is not only ine cient and unnecessary, but also unpro table. The cost-minimizing plan would have sustained the generous plan and lowered total compensation cost. 26 This underscores the fact that ELS is sub-optimal from both the rm s perspective and from society s. Despite this nding, it is not necessarily the case that more employees receive generous coverage under the cost-minimization rule than under the ELS rule. Further, the cost-minimization approach is not necessarily superior from a welfare perspective. When a cost-minimizing employer chooses to o er the generous plan, it does so in order to lower its compensation cost, not for any e ciency reasons. To shed further light on this issue, for an interior solution under the ELS rule, the healthiest worker in the generous plan, type c ELS g, satis es: Z g c ELS g = (1 t) c g f (c) c 1 F (c g ) dc Z cg c m! c f (c) F (c g ) dc. (11) Comparing (11) and (10) illustrates why, in general, c g and c ELS g cannot be ordered independent of the distribution of health types and the shape of g (c). According to (11), the ELS rule sets the marginal worker s willingness to pay for the generous plan equal to the di erence in the average cost of the two plans. The right-hand side of (10) di ers from the right-hand side of (11) in that the cost-minimizing employer cares about the actual cost of the marginal worker. Since the di erence in the cost of the marginal worker is less than the di erence in the average cost of the two plans, the right-hand side of (10) is smaller than the right-hand side of (11). All else equal, this leads the cost-minimizing employer to put more workers in the generous plan. However, the left-hand sides of (11) and (10) di er as well. Since the elasticity of willingness to pay for generous coverage is 25 Note that viability of the generous plan depends on the distribution of health types under the ELS rule but not under the cost-minimization rule. 26 To be fair, the ELS approach also has strong advantages which do not emerge in this model. Making employees pay the entire premium increase when choosing a more expensive health plan gives them strong incentives to choose less expensive plans, and this increased price sensitivity will increase insurers incentives to lower their prices (Enthoven 1988; 2003). Indeed, in their discussion of the Harvard case Cutler and Reber (1998) argue that increased plan-side competition resulting from adoption of the ELS rule reduced Harvard s premiums by 5 to 8 percent. 19

21 negative, the left-hand side of (10) is less than the left-hand side of (11). Taken in isolation, this induces the cost-minimizing employer to put fewer people in the generous plan than the ELS rule. Further, this e ect is more pronounced when the willingness to pay is relatively inelastic (i.e., j"j is small). Since these two e ects push in opposite directions, whether cost-minimization or ELS is more e cient (i.e., which rule comes closer to the socially optimal rule, (2)) cannot be determined without knowing more about the distribution of health types and the shape of g (c). The ELS rule is just one rule employers use. Other rules are less susceptible to adverse selection spirals. For example, Cutler and Zeckhauser (1998) contrast Harvard s practices with those used by the Group Insurance Commission of Massachusetts (GIC), which designs the health insurance options for state employees health bene ts. By Massachusetts law, the state must use a fairly generous EPP rule. Since employees out-of-pocket contribution is only a small fraction of the di erence in the cost of the two plans, this tends to dampen the e ects of adverse selection. Consequently, it is more likely that the generous plan remains viable under such a rule. Indeed, Cutler and Zeckhauser explain that, because of its contribution rule and other cost-containment measures, the GIC has controlled the e ects of adverse selection on its most generous plan viable. 6 Discussion This paper has considered an employer with the goal of minimizing its total compensation cost and has argued that in setting the surcharge for generous coverage, the employer should act as a monopolist who sells health insurance upgrades to its employees. At the most general level, this principle advocates a change in approach to the bene ts design problem: employers primary focus should be on managing self-selection in order to minimize total compensation cost. 27 Such an approach can ameliorate common bene ts-related problems employers face. For example, by explicitly considering the cost e ects of di ering selection, the employer controls it, reducing the likelihood and severity of unintended adverse selection and premium spirals. The theory of rm behavior laid out in this paper provides a basis for thinking about how cost-minimizing employers will react to policy interventions such as changes in the tax code or the 27 In fact, even when the employer has some other goal, we would argue that it could more easily achieve it by focusing on choosing the surcharge that induces its desired behavior rather than adhering to some rule that leaves the surcharge endogenously determined by employees choices. 20